கணிதம் உங்களுக்காக - 23 - கான்ஸெப்ட்கள்

1. Equation of an ellipse in standard form: (x² / a²) + (y² / b²) = 1 (where a is the semi-major axis and b is the semi-minor axis)

2. Equation of a hyperbola in standard form: (x² / a²) - (y² / b²) = 1 (horizontal) or (y² / a²) - (x² / b²) = 1 (vertical)

3. Equation of a line in standard form: Ax + By + C = 0

4. Distance from a point to a plane in 3D: d = |Ax₁ + By₁ + Cz₁ + D| / √(A² + B² + C²)

5. Equation of a sphere in standard form: (x - h)² + (y - k)² + (z - l)² = r² (where (h, k, l) is the center and r is the radius)

6. Distance between two points in 3D: d = √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)

7. Midpoint of a line segment in 3D: M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)

8. Volume of a regular polyhedron: V = (1/6) a³√(2/3)[5n(3 + 2√5)] (where a is the side length and n is the number of faces)

9. Surface area of a regular polyhedron: SA = n a²√3 (where a is the side length and n is the number of faces)

10. Volume of a tetrahedron: V = (1/6)|a · (b × c)| (where a, b, and c are vectors representing the edges)

11. Surface area of a tetrahedron: SA = √3 a² (where a is the side length)

12. Volume of an oblique cylinder: V = πr²h

13. Surface area of an oblique cylinder: SA = 2πrh + 2πr²

14. Volume of a frustum of a pyramid: V = (1/3)h(A₁ + A₂ + √(A₁A₂)) (where h is the height, A₁ is the area of the top base, and A₂ is the area of the bottom base)

15. Surface area of a frustum of a pyramid: SA = A₁ + A₂ + (1/2)Pℓ (where P is the perimeter of the base and ℓ is the slant height)

16. Conversion from Cartesian to polar coordinates: r = √(x² + y²), θ = tan⁻¹(y / x)

17. Conversion from polar to Cartesian coordinates: x = r cos(θ), y = r sin(θ)

18. Equation of a spiral: r = a + bθ (where a and b are constants)

19. Equation of a limaçon: r = a ± b cos(θ) or r = a ± b sin(θ)

20. Equation of a rose curve: r = a cos(kθ) or r = a sin(kθ)

21. Product-to-Sum Formulas:

    - sin(A) sin(B) = ½ [cos(A - B) - cos(A + B)]

    - cos(A) cos(B) = ½ [cos(A + B) + cos(A - B)]

    - sin(A) cos(B) = ½ [sin(A + B) + sin(A - B)]

22. Sum-to-Product Formulas:

    - sin(A) ± sin(B) = 2 sin((A ± B) / 2) cos((A ∓ B) / 2)

    - cos(A) + cos(B) = 2 cos((A + B) / 2) cos((A - B) / 2)

    - cos(A) - cos(B) = -2 sin((A + B) / 2) sin((A - B) / 2)

23. Half-Angle Formulas:

    - sin²(A/2) = (1 - cosA) / 2

    - cos²(A/2) = (1 + cosA) / 2

    - tan²(A/2) = (1 - cosA) / (1 + cosA)

24. Double Angle Formulas:

    - sin(2A) = 2 sinA cosA

    - cos(2A) = cos²A - sin²A = 2 cos²A - 1 = 1 - 2 sin²A

    - tan(2A) = 2 tanA / (1 - tan²A)

25. Triple Angle Formulas:

    - sin(3A) = 3 sinA - 4 sin³A

    - cos(3A) = 4 cos³A - 3 cosA

    - tan(3A) = (3 tanA - tan³A) / (1 - 3 tan²A)

26. Dot product: A · B = A₁B₁ + A₂B₂ + A₃B₃

27. Vector cross product: A × B = (A₂B₃ - A₃B₂)i + (A₃B₁ - A₁B₃)j + (A₁B₂ - A₂B₁)k

28. Magnitude of a vector in 3D: |A| = √(A₁² + A₂² + A₃²)

29. Scalar projection of vector A onto vector B: proj_B(A) = (A · B) / |B|

30. Vector projection of vector A onto vector B: Proj_B(A) = ((A · B) / |B|²) B

31. Angle between two vectors: cos(θ) = (A · B) / (|A||B|)

32. Distance between two skew lines: d = |(A₁ - A₂) · (B₁ × B₂)| / |B₁ × B₂| (where A₁ and A₂ are points on the lines, and B₁ and B₂ are direction vectors)

33. Power of a Point Theorem: If a point P lies outside a circle and two secant lines PA and PB intersect the circle at A and B, then PA × PB = PC × PD (where C and D are points of intersection)

34. Segment lengths in a circle: If two chords AB and CD intersect at point P inside the circle, then PA × PB = PC × PD.

35. Length of a tangent segment: If a tangent from an external point P touches the circle at T, then PT² = PA × PB (where A and B are points of intersection of a secant through P)

36. Length of an arc: L = rθ (where r is the radius and θ is the central angle in radians)

37. Area of a sector: A = 1/2r²θ (where r is the radius and θ is the central angle in radians)

38. Segment area: A = 1/2r²(θ - sinθ)

39. Length of the altitude of a right triangle: h = (a * b) / c (where a and b are the legs and c is the hypotenuse)

40. Radius of the circumscribed circle of a triangle: R = (abc) / (4A) (where a, b, and c are the side lengths and A is the area of the triangle)

41. Length of the median of a trapezoid: m = (a + b) / 2 (where a and b are the lengths of the parallel sides)

42. Length of the diagonal of a rectangle: d = √(a² + b²) (where a and b are the side lengths)

43. Length of the diagonal of a parallelogram: d = √(a² + b² + 2ab cos(θ)) (where a and b are the side lengths and θ is the angle between the sides)